Computer Simulations, Nucleation Rate Predictions and Scaling

Barbara N. Hale

Physics Department, University of Missouri-Rolla, Rolla, MO, 65409USA

Abstract. Computer simulations of atomic and molecular systems relevant to the prediction of nucleation rates have focused primarily on two approaches: (1) calculation of the free energy of formation of the critical embryo (or equivalently n-cluster probability ratios); and (2) time dependent simulation of embryo formation in a dense parent phase. Excepting a few scope- limited quantum mechanical treatments both these approaches use some form of effective atom- atom pair potentials. Successful nucleation rate prediction demands adequate treatment of entropic effects (configurational sampling) and hence all these approaches employ microscopic statistical mechanics. In Monte Carlo (MC) approaches particle configurations are sampled with a Boltzmann weighting subject to thermodynamic constraints such as constant T, N or  V or P. In molecular dynamics (MD) inter-atomic forces accelerate the atoms along time trajectories subject to system constraints. Since the mid-1990s, sampling efficiency algorithms, steadily increasing computing power and the ingenuity of young scientists have turned this field into an impressive statistical mechanical study of many body classical systems. The intent of this paper is to generate an initial literature data base relevant to these “nucleation” simulations, to gather the nucleation rate predictions and experimental data into unifying plots and to examine the implications of the scaled supersaturation in comparing the experimental data with the simulation predictions. In particular, homogeneous vapor-to-liquid nucleation rates estimated via MC and MD simulations are compared with experimental rate data for argon and for water.

Key Words: Homogeneous nucleation, computer simulations, Monte Carlo, molecular dynamics, effective pair potentials, density functional theory, nucleation rate, work of formation, free energy of formation, critical cluster, vapor-to-liquid phase transitions, liquid-to-solid phase transitions, critical nucleus, small cluster properties, cluster definition, microscopic many body interactions, thermodynamic constraints, scaled nucleation rate, scaled temperature dependence, scaling law, effective surface tension, argon, water, TIP4P, Lennard-Jones, scaled supersaturation

THE FIRST COMPUTER SIMULATIONS

This summary is intended for the nucleation community. The whole of microscopic simulation literature (much of which has relevance to nucleating systems) is immense. Many of the computer simulations do not predict a nucleation rate, but rather demonstrate techniques and test inter-atomic potentials for their applicability to real systems. Because it is not possible to present all the nucleation-directed computer simulation literature in this article a web site has been set up at http://web.umr.edu/~hale/nucleation.html for those interested in more details. The abstracts for all the articles are included. A more extensive discussion of the simulational details for the Monte Carlo and molecular dynamic methods is given by H. Vehkamaki in this volume. The history of these simulations begins in the late 1960s and 1970s, when simply to carry out a few hundred thousand step Monte Carlo simulation was a feat. To these early workers1-41 much is owed: Metropolis1, Alder and Wainwright2, Wood and Parker3, Hoover and Ree4, Hansen and Verlet5, McDonald6, Norman and Filinov7, Rahman8, Johannesson9, Stillinger et al.10, McGinty11, Adams and Jackson12, Lee, Barker and Abraham13, Robertson and Pound14, Stauffer et al.15, and Briant and Burton16, to name a few. While many were simulating lattice gases15,24,28,32,38 or argon Lennard-Jones systems5-8,11,13,18,23,26,27,29,31,33,34,39-41 because of the generally accepted potential and enhanced computational speed, a few were attempting water systems with a variety of potentials.10,16,17,19,20,22,30,35,36 The molecular dynamics simulations of liquid water by Stillinger10 and the water-water potentials developed for these simulations were especially important to subsequent studies focused on the nucleation of atmospheric water and ice35.

DATA ANALYSIS AND THE SCALED SUPERSATURATION

In 1980s homogeneous nucleation experiments42-45 began to provide more detailed information about the dependence of the nucleation rate, J, on temperature, T, and supersaturation ratio, S. The data of Schmitt et al.44 for toluene is plotted in Fig. J (O) J (+) 1a. exp scaled Jexp (O) Jscaled (+) 259K 259K 233K Figure 1. The homogeneous toluene nucleation 217K 233K rate data of Schmitt et al 44 plotted a) vs. lnS 217K and b) vs. the scaled supersaturation 46, ln S / 3/2 3/2 [Tc/T -1] . The constant Co = [Tc/240 -1] .

Fig. 1b shows logJ plotted as a function 46,47 of the scaled supersaturation , lnSscaled, 3/2 where lnSscaled = lnS / [Tc/T -1] . This scaled supersaturation is similar to that first proposed by Binder48 and is taken from the scaling law46 we developed in 1986. One can see the effect of the 4 a) scaling as it collapses all the 4 b) Schmitt et al. toluene data “temperature data lines” in Fig. 1a onto a 3 Schmitt et al. toluene data

single line. Assuming that the data holds no ) 3 1 - s ) unexpected surprises, this effect is one that should not be missed. It suggests the 3 - 1 2 - s m

specific combination of S and T upon which the nucleation rate data depends: J exp = J( 3 c -

3/2 lnS / [Tc/T -1] ). m This method of plotting the data permits a check for consistency (and J c (

scaling) without resorting to ad hoc parameters. The data for nonane of Adams et al. o l ( 1 (Fig. 2) also exhibits this scaling, though to a less satisfactory degree. g o l 0 0 -1 -1 2 3 4 2 l3nS 4 3/2 Co lnS/[Tc/T-1] J (O) J (+) exp scaled Jexp (O) Jscaled (+) 259K 259K 233K 233K 217K 217K

6 6 a) b) Adams et al. nonane data Adams et al. nonane data 5 5 ) 1 ) - 1 s - 3 s - 4 3

- 4 m c m

3 J g ( 3 o g l o l 2 2 Figure 2. The homogeneous nonane nucleation rate data of Adams et al 42 plotted a) vs. lnS and b) vs. 46 3/2 3/2 1 the scaled supersaturation , ln S / [Tc/T -1] . The constant, Co = [Tc/240 -1] . 2 3 4 5 1 2 3 4 5 Probablyl nmostS surprising is the scaling of the homogeneous nucleation rate data for 3/2 water, a polar substance of considerable complexity. Co lnS/[ T cThis/T-1 ]is shown in Fig. 3. for the data of Wölk and Strey45.

a) b) 10 260 K 250 K 10 Wolk and Strey H2O data 240 K 230 K 240 K 230 K

220 K ]

) 1 1 - - c c e e s

s / 3 8 - 8 3 - m m c ( c / 255 K

6 o 6 l

4 4 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3/2 lnS Co lnS / [Tc/T -1]

Figure 3. The homogeneous water nucleation rate data of water, Wölk and Strey45 plotted vs. a) lnS and 46 3/2 3/2 b) the scaled supersaturation , ln S / [Tc/T -1] . The constant, Co = [Tc/240 -1] .

Actually it has been known for some time that the classical Becker-Döring nucleation rate model has the wrong temperature dependence. Recently, Wölk et al.49, developed an empirical temperature fitting function which alters the Becker-Döring nucleation rate, JBD, to agree with the experimental water data. It was noted that upon this 46,47 conversion, JBD becomes nearly identical to the scaled nucleation rate model . It was a demonstration of this which motivated the scaled supersaturation plots in Figs. 1-3.47 EXPERIMENTS AND SIMULATIONS: WATER The scaling of J permits comparison of a range of numerical simulation data with experimental data taken under different conditions. Molecular dynamics simulations of vapor-to-liquid systems are often carried out in dense vapor systems at relatively high temperatures,50-54 where experimental data is lacking. In 1998 Yasuoka and Matsumoto52 carried out a large scale molecular dynamics simulation on a system of 5000 carrier gas atoms and 5000 water molecules interacting via the TIP4P water- water potential. The system temperature was themostated at 350K via the carrier gas atoms after the quench. Vapor depletion reduced the initial supersaturation of 14.6 to an average value near S = 7.3, corresponding to the observed nucleation rate of 9.62 1026 cm-3s-1 ; the critical nucleus was in the range of 30- 45 water molecules. Although the nucleation rate is larger than any observed, it can be compared roughly with the nozzle data of Heath et al.54 and the expansion chamber data of Wölk and Strey45 on a 26 3 2 scaled plot. This is shown in Fig. 4., where log[10 / J] is plotted vs. [Tc//T -1] / (ln S) . ]

H O: Miller et al.

s 3 - 2 0 H O: Wolk and Strey m 2 c

0 MC TIP4P 1

Vehkamaki /

Figure 4. J

The log

D O, H O 2 2 26 [ [10 /J ] Wyslouzil et al. g vs. [Tc/T

o 3 l -1] / (ln

- 0S) 2 for the MD TIP4P: Yasuoka et al. water data of Miller et al .55, the fast 0 10 20 30 expansion chamber 3 2 data of C'o [Tc/T -1] / (lnS) Wölk and Strey45, the nozzle data of Wyslouzil et al.56-58, the MD simulation of TIP4P water by Yasuoka and 52 112 Matsumoto , and the MC TIP4P calculations of Merikanto et al. The constant, C’o = [16 /ln10 with he dashed line is the scaled model prediction46 using see also TIP4P MCDS model simulations of Hale and DiMattio87. The molecular dynamics result for J does not fall on the line predicted by the experimental data. However, the simulation assumes a TIP4P model for water which does not reproduce the experimental surface tension. The Monte Carlo simulations of Merikanto et al.112 using TIP4P water appear to have the right temperature dependence but the predicted nucleation rates are too large by a factor of about 104. EXPERIMENTS AND SIMULATIONS: ARGON A large number of computer simulations beginning in the1990s have used the Monte Carlo method to sample the configuration space of atoms (or molecules) in small clusters.59-95 Several techniques can be used to determine the critical cluster size, n*, and calculate n-cluster energies of formation. One can determine a steady state 1 2n* 1 nucleation rate from   where Jn is the equilibrium forward rate for size n; or one J n1 Jn can insert the calculated energy of formation and vapor pressure into the classical model. Most of these simulations have been focused on the rare-gas, Lennard-Jones (LJ) system. ComputationallyArgon Rate Da thista a hasnd considerableLJ Simulat advantage.ions But experimental argon

1 - s 3 - m c

6 30 2

g 10 o l

-10 0 10 20 30 40 50 3 2 C'o [Tc/T -1] / (lnS)

26 3 2 78 Figure 5. The log [10 /J ] vs. [Tc/T -1] / (ln S) for the experimental argon data of Zahoransky (▽), Fladerer113 (+), Wu et al.114 (X), Matthew and Steinwandel115 (△), and Stein116 (). The theoretical predictions are: Yasuoka and Matsumoto106 (), ten Wolde and Frenkel73, 82 using S = 1.53 (), Hale68 for S and T corresponding to Fladerer data (), Senger et al.81 (), Oh and Zeng86 (), Chen et al.90 (), and CNT with Zeldovich factor for S and T corresponding to Fladerer data (solid line). The C’o = [16/ln(10) with . he scaled model prediction with short-dashed line) passes through the Fladerer data. The long-dashed line shows the scaled model prediction46 with  the bulk experimental excess surface entropy/k per molecule, as determined from the surface tension. vapor-to-liquid nucleation rates have proven more difficult to measure. In Fig. 5 is a scaled supersaturation plot of simulation results for Lennard-Jones compared with some of the experimental argon rate data. The experiments generally quote data for “onset” nucleation conditions. For all but one set of the experimental data a rate of 1 cm-3s-1 is used in making the plot. For the data of Fladerer113 an approximate rate of 107  2 cm-3s-1 is used. At large scaled supersaturations (small values of the abscissa in Fig. 5) the Monte Carlo simulations of Senger et al.81, Oh and Zeng86, Chen et al.90, and Hale68 are at temperatures near 85K and S = 5. These data predict nucleation rates of the order of 1017- 1022 cm-3s-1. The prediction of ten Wolde and Frenkel73, 82 at T = 88.9K and S = 1.53 is 105 cm-3s-1. This simulation uses a truncated LJ potential with surface tension about 30% smaller than the untruncated LJ potential. The scaled supersaturation for this point was determined 106 from reduced Tc =1.085 and T = 0.741. The MD result of Yasuoka and Matsumoto is larger (1027 cm-3s-1) and is at T = 80K, S=7.2. The values of S for the theoretical data o are calculated from the ratio of Lennard-Jones potential model vapor densities, v /v , o where v is the vapor density in equilibrium with the liquid. Unfortunately, no experimental data exist at the large scaled supersaturations. In the smaller scaled supersaturation region of the experimental data, the MC simulations68 and the classical nucleation theory (CNT) give rates of 10-10 cm-3s-1 and 10-13 cm-3s-1 , respectively --- about a factor of 10-17 to 10-20 smaller than the “onset” nucleation indicated by the experimental data. These data are at T =50-59K and S= 8-16. In the small scaled supersaturation region the energies of formation are large; and for large scaled supersaturations the energies of formation are small, giving larger rates. The scaled model46 with  = 1.5 (dashed line) passes roughly through the Fladerer data. This value of  (the excess surface entropy/k per molecule) is about 30% smaller than the bulk -1 -2/3 experimental value of 2.1 as determined from –k liq [/T], where  is the bulk surface tension and liq is the liquid number density. The prediction of the scaled model with  = 2.1 is indicated in Fig. 5 by the long-dashed line. Clearly for argon, challenges remain for both theorist and experimentalist.

A number of studies have used the density functional approach.96-98 Zeng and Oxtoby97 note that their density functional treatment of the Lennard-Jones system gives good agreement with the experimental gas to liquid nonane data. Their non-classical nucleation rate, JNCL, is about four orders of magnitude larger than the classical nucleation theory (JCNT). Computer simulations of liquid-solid homogeneous nucleation99-111 are also noted here because of the great interest the atmospheric community has in ice formation.

COMMENTS AND CONCLUSIONS

The simulation results have made great progress, particularly in the last six years. Experimental determination of homogeneous nucleation rates remains a challenge, especially in the case of argon for which the simulational treatment holds the most advantage. There appears to be a general need for those reporting simulation results to include predicted nucleation rates, so that the many calculations can be tied together and compared with the experimental data. At the high temperatures and large S, the non-ideality of the vapor needs to be taken into account. Finally, it is suggested that the scaling properties of the nucleation rate might prove helpful to the theorists in their simulations as well as to the experimentalists in probing the temperature dependence of the data. References

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